Internal problem ID [15522]
Internal file name [OUTPUT/15523_Friday_May_10_2024_05_47_33_PM_7763517/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 3 (Systems of differential equations). Section 21. Finding integrable
combinations. Exercises page 219
Problem number: 792.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs" Unable to solve or complete the solution.
Solve \begin {align*} x^{\prime }\left (t \right )&=\frac {{\mathrm e}^{-t}}{y \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {{\mathrm e}^{-t}}{x \left (t \right )} \end {align*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 52
dsolve([exp(t)*diff(x(t),t)=1/y(t),exp(t)*diff(y(t),t)=1/x(t)],singsol=all)
\begin{align*} \left \{x \left (t \right ) &= \sqrt {-2 \,{\mathrm e}^{-t} c_{1} +2 c_{2}}, x \left (t \right ) &= -\sqrt {-2 \,{\mathrm e}^{-t} c_{1} +2 c_{2}}\right \} \\ \left \{y \left (t \right ) &= \frac {{\mathrm e}^{-t}}{\frac {d}{d t}x \left (t \right )}\right \} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 125
DSolve[{Exp[t]*x'[t]==1/y[t],Exp[t]*y'[t]==1/x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -\sqrt {2} \sqrt {c_1} \sqrt {-e^{-t}+c_1 c_2} \\ x(t)\to -\frac {\sqrt {-2 e^{-t}+2 c_1 c_2}}{\sqrt {c_1}} \\ y(t)\to \sqrt {2} \sqrt {c_1} \sqrt {-e^{-t}+c_1 c_2} \\ x(t)\to \frac {\sqrt {-2 e^{-t}+2 c_1 c_2}}{\sqrt {c_1}} \\ \end{align*}